Step Response of Commensurate Fractional Lowpass Pseudo-Biquad: Critical Damping

In the paper, a comparison is made between the step responses of the classical integer-order biquad with the transfer function
$((s/\omega_{0})^{2}+(s/\omega_{0})/Q+1)^{-1}$
, where
$\omega_{0}$
and
$Q$
are the characteristic frequency and quality factor, and the commensurate fractional pseudo-biquad with the transfer function
$((s/\omega_{0})^{2\alpha}+(s/\omega_{0})^{\alpha}/Q+1)^{-1},\ 0 < \alpha\leq 1$
. While the classical biquad experiences the fastest response for critical damping when
$Q=0.5$
, a similar response can be observed for the fractional circuit, but for larger values of
$Q$
depending on the
$\alpha$
parameter. For
$\alpha < 1$
, the response then settles faster than for the classical filter. Coupling conditions between the parameters
$\alpha$
and
$Q$
are found that lead to the so-called pseudo-critical damping and critical underdamping.